(1 + ε)-Approximation for Facility Location in Data Streams∗
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Bibliographic record
Abstract
We consider the Euclidean facility location problem with uni-form opening cost. In this problem, we are given a set of n points P ⊆ R2 and an opening cost f ∈ R+, and we want to find a set of facilities F ⊆ R2 that minimizes f · |F |+ p∈P min q∈F d(p, q), where d(p, q) is the Euclidean distance between p and q. We obtain two main results: • A (1 + ε)-approximation algorithm with running time O(n log2 n log log n) for constant ε, • The first (1 + ε)-approximation algorithm for the cost of the facility location problem for dynamic geometric data streams, i.e., when the stream consists of insert and delete operations of points from a discrete space {1,...,∆}2. The streaming algorithm uses log ∆ ε)O(1) space. Our PTAS is significantly faster than any previously known (1 + ε)-approximation algorithm for the problem, and is also relatively simple. Our algorithm for dynamic geometric data streams is the first (1 + ε)-approximation algorithm for the cost of the facility location problem with polylogarithmic space, and it resolves an open problem in the streaming area. Both algorithms are based on a novel and simple decompo-sition of an input point set P into small subsets Pi, such that: • the cost of solving the facility location problem for each Pi is small (which means that one needs to open only a small, polylogarithmic number of facilities), • ∑i OPT(Pi) ≤ (1 + ε) ·OPT(P), where for a point set P, OPT(P) denotes the cost of an optimal solution for P. ∗Research partially supported by the EU within the 7th Framework
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.002 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it